There is mounting evidence that compact elliptical galaxies (CEGs) are local analogs of the high-redshift red nuggets thought to represent progenitors of todays early-type galaxies (ETGs). We report the discovery of extended X-ray emission from a hot interstellar / intragroup medium in two CEGs, Mrk 1216 and PGC 032873, using shallow archival Chandra observations. We find that PGC 032873 has an average gas temperature $k_BT=0.67pm 0.06$ keV within a radius of 15 kpc, and a luminosity $L_{rm x} = (1.8pm 0.2)times 10^{41}$ erg s$^{-1}$ within a radius of 100kpc. For Mrk 1216, which is closer and more luminous $[L_{rm x}(rm <100~kpc) = (12.1pm 1.9)times 10^{41}$ erg s$^{-1}]$, we performed a spatially resolved spectral analysis in 7 annuli out to a radius of 73 kpc. Using an entropy-based hydrostatic equilibrium (HE) procedure, we obtain a good constraint on the $H$-band stellar mass-to-light ratio, $M_{rm stars}/L_H=1.33pm 0.21$ solar, in good agreement with stellar dynamical (SD) studies, which supports the HE approximation. We obtain a density slope $2.22pm 0.08$ within $R_e$ consistent with other CEGs and normal local ETGs, while the dark matter (DM) fraction within $R_e$, $f_{rm DM}=0.20pm 0.07$, is similar to local ETGs. We place a constraint on the SMBH mass, $M_{rm BH} = (5pm 4)times 10^{9}, M_{odot}$, with a 90% upper limit of $M_{rm BH} = 1.4times 10^{10}, M_{odot}$, consistent with a recent SD measurement. We obtain a halo concentration $(c_{200}=17.5pm 6.7)$ and mass [$M_{200} = (9.6pm 3.7)times 10^{12}, M_{odot}$], where $c_{200}$ exceeds the mean $Lambda$CDM value ($approx 7$), consistent with a system that formed earlier than the general halo population. We suggest that these galaxies, which reside in group-scale halos, should be classified as fossil groups. (Abridged)
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